Update: Bezos' recent Tweeted update on the 10,000 year clock. Also see The Verge's Construction begins on Jeff Bezos’ $42 million 10,000-year clock
The discussions associated with the question Can an artificial satellite stay in orbit forever? and its answers suddenly reminded me of the 10,000 year clock project. Take a moment to look there now.
I also thought about the passive mechanical structures described in this answer — basically bi-metallic strips that open/close shutters, but their designed behavior reacting to temperature distributions inside is actually pretty complicated.
An Earth-orbiting satellite that needs to remain in orbit for a long time should at perigee be high enough to avoid essentially all atmospheric drag, but not too high in order to at least reduce the gravitational perturbations from Moon, Sun, etc.
It could passively absorb thermal energy from the thermal difference between Sun and space, and store it through springs and gears and latches. Then, maybe once a year it would have enough power to produce a signal - by radio, or light. 10,000 year electronics might not be impossible if they are low tech (discrete devices) and efficient, and one "beep" per year may not be very demanding. Whatever it does should be fairly limited - not impose itself on those not wanting to see it specifically.
As a backup, if the active electronic "beep" fails, I suppose it could mechanically just change reflectivity, and "suddenly appear" once a year. Large surface area can be obtained in various ways — unrolling something then rolling it back up just for example. And it doesn't have to launch next month — it could be a development project (like the 10,000 year clock). Once it's up there, it wouldn't require any further management from the ground.
Calculating the orbital mechanics would be a substantial project, so I'm not asking here for an orbit that can last 10,000 years. If there are calculations showing it can not exist, please post the link or calculations! But otherwise let's avoid the "Orbital Mechanical Opinions".
What else besides orbital mechanics is there that would pose the most difficult challenges to making an artificial satellite stay in an orbit around the earth for 10,000 years, and "beep" or change appearance once a year?